Theory of Probability and Mathematical Statistics
Parisian times for linear diffusions
Christophe Profeta
Link
Abstract: We compute the joint distribution of the first times a linear diffusion makes an excursion longer than some given duration above (resp. below) some fixed level. In the literature, such stopping times have been introduced and studied in the framework of Parisian barrier options, mainly in the case of Brownian motion with drift. We also exhibit several independence properties, and provide some formulae for the associated ruin probabilities.
Keywords: Diffusion theory, excursion theory
Bibliography: J. H. M. Anderluh and J. A. M. van der Weide, Double-sided Parisian option pricing, Finance Stoch. 13 (2009), no. 2, 205–238. MR 2482052, DOI 10.1007/s00780-009-0090-3
Andrei N. Borodin and Paavo Salminen, Handbook of Brownian motion—facts and formulae, 2nd ed., Probability and its Applications, Birkhäuser Verlag, Basel, 2002. MR 1912205, DOI 10.1007/978-3-0348-8163-0
Marc Chesney, Monique Jeanblanc-Picqué, and Marc Yor, Brownian excursions and Parisian barrier options, Adv. in Appl. Probab. 29 (1997), no. 1, 165–184. MR 1432935, DOI 10.2307/1427865
Angelos Dassios and Jia Wei Lim, An analytical solution for the two-sided Parisian stopping time, its asymptotics, and the pricing of Parisian options, Math. Finance 27 (2017), no. 2, 604–620. MR 3635299, DOI 10.1111/mafi.12091
Angelos Dassios and Jia Wei Lim, Parisian option pricing: a recursive solution for the density of the Parisian stopping time, SIAM J. Financial Math. 4 (2013), no. 1, 599–615. MR 3090647, DOI 10.1137/120875466
Angelos Dassios, Jia Wei Lim, and Yan Qu, Azéma martingales for Bessel and CIR processes and the pricing of Parisian zero-coupon bonds, Math. Finance 30 (2020), no. 4, 1497–1526. MR 4154777, DOI 10.1111/mafi.12248
Angelos Dassios and Shanle Wu, Double-barrier Parisian options, J. Appl. Probab. 48 (2011), no. 1, 1–20. MR 2809883, DOI 10.1239/jap/1300198132
Angelos Dassios and Shanle Wu, Perturbed Brownian motion and its application to Parisian option pricing, Finance Stoch. 14 (2010), no. 3, 473–494. MR 2670422, DOI 10.1007/s00780-009-0113-0
Van E. Wood, Table errata: Tables of integral transforms, Vol. I, II (McGraw-Hill, New York, 1954) by A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Math. Comp. 23 (1969), no. 106, 468. MR 415977, DOI 10.1090/S0025-5718-1969-0415977-9
R. K. Getoor, Excursions of a Markov process, Ann. Probab. 7 (1979), no. 2, 244–266. MR 525052, DOI 10.1214/aop/1176995086
Kiyosi Itô and Henry P. McKean Jr., Diffusion processes and their sample paths, Die Grundlehren der mathematischen Wissenschaften, Band 125, Springer-Verlag, Berlin-New York, 1974. Second printing, corrected. MR 345224
Olav Kallenberg, Foundations of modern probability, Probability Theory and Stochastic Modelling, vol. 99, Springer, Cham, [2021] ©2021. Third edition [of 1464694]. MR 4226142, DOI 10.1007/978-3-030-61871-1
S. Kotani and S. Watanabe, Kreĭn’s spectral theory of strings and generalized diffusion processes, Functional analysis in Markov processes (Katata/Kyoto, 1981) Lecture Notes in Math., vol. 923, Springer, Berlin-New York, 1982, pp. 235–259. MR 661628
Céline Labart and Jérôme Lelong, Pricing double barrier Parisian options using Laplace transforms, Int. J. Theor. Appl. Finance 12 (2009), no. 1, 19–44. MR 2517580, DOI 10.1142/S0219024909005154
Ronnie Loeffen, Irmina Czarna, and Zbigniew Palmowski, Parisian ruin probability for spectrally negative Lévy processes, Bernoulli 19 (2013), no. 2, 599–609. MR 3037165, DOI 10.3150/11-BEJ404
Jim Pitman and Marc Yor, Bessel processes and infinitely divisible laws, Stochastic integrals (Proc. Sympos., Univ. Durham, Durham, 1980) Lecture Notes in Math., vol. 851, Springer, Berlin, 1981, pp. 285–370. MR 620995
Jim Pitman and Marc Yor, Hitting, occupation and inverse local times of one-dimensional diffusions: martingale and excursion approaches, Bernoulli 9 (2003), no. 1, 1–24. MR 1963670, DOI 10.3150/bj/1068129008
Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999. MR 1725357, DOI 10.1007/978-3-662-06400-9
Shinzo Watanabe, On time inversion of one-dimensional diffusion processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 31 (1974/75), 115–124. MR 365731, DOI 10.1007/BF00539436
